Abstract
Let be a closed oriented surface of negative Euler characteristic and a complete contractible Riemannian manifold. A Fuchsian representation strictly dominates a representation if there exists a –equivariant map from to that is –Lipschitz for some . In a previous paper by Deroin and Tholozan, the authors construct a map from the Teichmüller space of the surface to itself and prove that, when has sectional curvature at most , the image of lies (almost always) in the domain of Fuchsian representations strictly dominating . Here we prove that is a homeomorphism. As a consequence, we are able to describe the topology of the space of pairs of representations from to with Fuchsian strictly dominating . In particular, we obtain that its connected components are classified by the Euler class of . The link with anti-de Sitter geometry comes from a theorem of Kassel, stating that those pairs parametrize deformation spaces of anti-de Sitter structures on closed –manifolds.
Citation
Nicolas Tholozan. "Dominating surface group representations and deforming closed anti-de Sitter $3$–manifolds." Geom. Topol. 21 (1) 193 - 214, 2017. https://doi.org/10.2140/gt.2017.21.193
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